Pacific
Journal of
Mathematics
APPROXIMATION BY NORMAL ELEMENTS WITH FINITE
SPECTRA IN C ∗ -ALGEBRAS OF REAL RANK ZERO
H UAXIN L IN
Volume 173
No. 2
April 1996
PACIFIC JOURNAL OF MATHEMATICS
Vol. 173, No. 2, 1996
APPROXIMATION BY NORMAL ELEMENTS WITH FINITE
SPECTRA IN C*-ALGEBRAS OF REAL RANK ZERO
HUAXIN LIN
We study the problem when a normal element in a C*algebra of real rank zero can be approximated by normal
elements with finite spectra. We show that all purely infinite simple C*-algebras, irrational rotation algebras and some
types of C*-algebras of inductive limit of the form C(X) <8> Mn
of real rank zero have the property weak (FN), i.e., a normal
element x can be approximated by normal elements with finite
spectra if and only if T(x) = 0 (λ — x G Inυo(A) for all λ ^ sp(x)).
For general C*-algebras with real rank zero, we show that a
normal element x with dim sp(x) < 1 can be approximated by
normal elements with finite spectra if and only if Γ(x) = 0.
One immediate application is that if A is a simple C*-algebra
with real rank zero which is an inductive limit of C*-algebras
of form C(Xn) <8> Λfm(n), where each Xn is a compact subset of
the plane, then A is an AF-algebra if and only if K\(A) = 0.
1. Introduction.
The notion of real rank for C*-algebras was introduced by L.G. Brown and
Gert K. Pedersen ([BP]). At present, it seems that it is the notion of real rank
zero that attracted most attention. (See [BBEK], [BDR], [BP], [BKR],
[CE], [E112], [EE],[G], [GL], [LZ], [Zhl-4], [Ph], [Lin4], [Lin5], and other
articles. When this revision is writing, many other articles on the subject
are appearing.) It turns out that the class of C*-algebras of real rank zero
is fairly large. In fact, from the remarkable work of G.A. Elliott ([E112]), for
any countable unperforated graded ordered group G with Riesz decomposition property, there exists a separable nuclear C*-algebra A of real rank zero
( and stable rank one) such that the graded ordered group (K0(A), Kι (A)) is
isomorphic to G. ( See [E113, 1.2 and 5.1] for the definition of unperforated
graded ordered group. Notice that Elliott's definition of unperforated is in
fact weaker than that of weakly unperforated in [B13, 6.7.1].) If A has real
rank zero, then A has the property (FS), i.e., every selfadjoint element can
be approximated by selfadjoint elements with finite spectra (see [BP, 2.6])
(and converse is also true). What about unitaries? Does real rank zero imply
the property (FU), i.e., every unitary can be approximated by unitaries with
443
444
HUAXIN LIN
finite spectra? One quickly realizes (for example, from Elliott's classification
theorem [E112]) that there are many C*-algebras with real rank zero have
nontrivial K\-groups. It is clear that those unitaries which are not connected
to the identity can not be approximated by unitaries with finite spectra. N.
C. Phillips introduced the notion of weak (FU), i.e., every unitary in the
connected component containing the identity can be approximated by unitaries with finite spectra ([Phi]). Many C*-algebras of real rank zero were
proved to have the property weak (FU) ([Phl-2], [GL]). Recently, we show
that, in fact, every C*-algebra with real rank zero has the property weak
(FU)([Lin5]). This result has many applications in the study of the unitary groups and KΊ-groups and is shown to have applications to C*-algebra
extension theory (see [Lin3], [Lin4] and Lemma 2.2). Next, of course, we
will consider general normal elements. At this point we would like to mention the BDF-theory. It is known that the Calkin algebra A ** B{H)/K,
where H is the separable, infinite dimensional Hilbert space and K is the
C*-algebra of compact operators, has real rank zero. One crucial result in
the BDF-theory is that a normal element x in the Calkin algebra can be approximated by normal elements with finite spectra if and only if the index of
rr, Γ(x) is zero. All of these previous results lead us to the following question:
Q Is it true that in a C*-algebra with real rank zero, a normal element
x can be approximated by normal elements with finite spectra if and only if
Γ(x) = 0 {the index Γ will be defined later)?
Our result for weak (FU) in [Lin5] shows that if we further assume that
sp(x) = S 1 , then the answer to Q is affirmative. The BDF-theory shows
that the answer to Q is affirmative if A is the Calkin algebra. Recent results
in [Lin4] show that for many other corona algebras the answer to Q is also
affirmative. If all of these give evidence that, in general, the answer to Q
should be positive, we would like to remind the readers that the question for
AF-algebras remains open, i.e., we do not know if every normal element in
an AF-algebra can be approximated by normal elements with finite spectra,
even though some progresses have been made (see [Lin6]).
In the present paper, we show that the answer to Q is affirmative for purely
infinite simple C*-algebras and for many other C*-algebra of real rank zero
such as the Bunce-Deddens algebras and irrational rotation algebras. For
general C*-algebras, we show that if we restrict to those normal elements x
with dim sp(#) < 1, the answer to Q is affirmative. But for more general
normal elements, we have to work in the A® K (see Theorem 3.13). While
the author still believes the answer to Q should be affirmative, Theorem 3.13
might be the best form we can have for the general case at present.
APPROXIMATION BY NORMAL ELEMENTS
445
An affirmative answer to Q has some interesting implications. Let X(n,ϊ)
be a sequence of compact subsets of the plane and A be a, C*-inductive limit
of C*-algebras of the form
Some necessary and sufficient conditions for A having real rank zero are given
in [BEEK]. Prom [E112], if every X(n,i) is either the unit circle or the unit
interval, then such C*-algebras of real rank zero are completely determined
by their Km. An immediate consequence of an affirmative answer to Q is
that A is in fact an AF-algebra if and only if Kλ (A) = 0. So in the case that
Ki (A) = 0, these algebras will be completely determined by their i^0-groups
([Elll]). It should not be hard, then, from an affirmative answer to Q, that
in fact these C*-algebras of real rank zero in general (without assuming their
AΊ-groups being trivial), are completely determined by their ^-groups. In
particular, these C*-algebras are included in [E112].
The paper is organized as follows: Section 2 gives a generalized notion
of index Γ for normal elements and some basic facts which we are needed
in subsequent sections. In Section 3, we present some technical approximation results in A ® K which are important for the rest of the paper. In
Section 4, we show that for purely infinite simple <7*-algebras, irrational rotation algebras, the Bunce-Deddens algebras and many other C*-algebras
with real rank zero, the answer to Q is affirmative. In Section 5, we show
that for general C*-algebras with real rank zero, a normal element x with
dim sp(x) < 1 can be approximated by normal elements with finite spectra
if and only if Γ(x) = 0. Finally, in Section 6, we give some applications. We
will show, for example, if A is a simple C*-algebra of real rank zero which
is an inductive limit of C*-algebras of the from C(Xn) <8> M m ( n ), where each
Xn is a contractible compact subset of the plane, then A is an AF-algebra.
The following notations are used throughout this paper.
Let A be a C*-algebra. We use the notation A** for the enveloping W*algebra. If p is an open projection (of A) in A**, Her(p) is the hereditary
C*-subalgebra pA**p Π A.
Let
Ax -> A2 -> -> An -»
be a sequence of C*-algebras and φn : An —> An+ι be the connecting homomorphisms. We will use the notation \iτn^(An,φn) for the C*-inductive
limit and φ^An) for the image of An in the inductive limit.
Added in proof: This paper was written in 1992. Since then there are
significant development in the study of C*-algebras of real rank. We would
only like to mention that, by the result that a pair of almost commuting self-
446
HUAXIN LIN
adjoint matrices is close to a pair of commuting selfadjoint matrices proved
by the author, every AF-algebras has (FN).
2. Preliminaries.
While 2.2 and 2.3 are certainly new, most results stated in this section are
either routine or easy consequences of some known facts. Since many of
these will be needed frequently in the subsequent sections, we present them
here for reader's convenience. Some sketch of proofs are also presented.
Definition 2.1.
Let A be a unital C*-algebra and X be a compact
Hausdorff space. Suppose that φ : C(X) —> A is a monomorphism and
φ* : Kι(C(X)) —> Kι(A) is the induced homomorphism. Let a b e a normal
element in A. Denote by B the C*-subalgebra of A generated by x and the
identity 1. Then B = C(X), where X = sp(α ). This gives a monomorphism
φ : C{X) -> A. We define
Γ(x)=ΓA(x)
= φm.
If A is not unital, we define T(x) = T^(x).
Suppose that A is unital. We denote the unitary group of A by U(A) and
the path connected component containing the identity by U0(A). There is
a homomorphism i from U(A)/U0(A) into Kχ(A). We will use the following
lemma which is an application of our result for weak (FU) ([Lin5]).
Lemma 2.2. Let A be a (unital) C*-algebra of real rank zero. Then the
map i : U(A)/U0{A) ->• Kλ(A) is injective.
Proof. Suppose that u G U(A) and V = diag(l, 1,..., 1,IA) is in U0(Mk(A))
for some integer k. It follows from [BP, 2.10] that Mk(A) has real rank
zero. So, by [Lin5], V can be approximated by unitaries in Mk(A) with
finite spectra. It follows from [Lin5, Lemma 3] that u can be approximated
by unitaries in A with finite spectra. Since unitaries with finite spectra are
connected by a path with the identity, we conclude that u G U0(A).
D
When X is a compact subset of the plane, it is well known that
Kx[C{X))^U{C{X))IUo{C{X)).
So Γ(x) is a homomorphism from U(C(sp(x)))/U0(C(sp(x)))
into
U(A)/U0(A), when the map i is injective, in particular, when A has real rank
zero. Let lnv(A) denote the group of invertible elements in A and Invo(A)
denote the path connected component of Inv(A) containing the identity. It
is well known that Inv(A)/Invo(A) = U(A)/U0(A). We also will use the notation π x (X) for the group Inv(C(X))/Invo(C(X))
^
U{C(X))/UO(C(X)).
APPROXIMATION BY NORMAL ELEMENTS
447
L e m m a 2.3. Let A be a (unital) C*algebra of real rank zero and let p G A
be a projection. Ifx G pAp and y G (1 — p)A(l —p) such that x + y G Invo(^4)
and y G Inv o ((l — p)A(l — p)), then x G Invo(pAp).
Proof Let u be the unitary part of the polar decomposition of x + y in A.
Then u G U0(A). It follows from [Lin5] that A has the property weak (FU).
So u can be approximated by unitaries in A with finite spectra. We can
write u — uλ + u2, where ux G U(pAp) and u2 G U((l — p)A(l — p)). Since
y G Inv o ((l - p)A{l - p ) ) , u 2 e ϋΌ((l -p)A(l - p ) ) . By [Lin5] again, u 2
can be approximated by unitaries in (1 — p)A(l — p) with finite spectra.
Then, from [Lin5, Lemma 3], we conclude that U\ can be approximated by
unitaries in pAp with finite spectra. Hence u G U0(pAp). This implies that
x G Invo(pAp).
D
Proposition 2.4. Ze£ ^4 δe a (unital) C*-algebra and let x G A be a normal
element. Then Γ(x) — 0 if and only if for any λ ^ sp(#), λ — x G Invo(A).
Or equivalently, the unitary part of the polar decomposition of λ — x is in
U0(A) for all λ^sp(x).
Proof. The "only if part is trivial from the definition of Γ. For the "if"
part, we know that π1(sp(x)) is the free abelian group with a generator for
each bounded component of C \ sp(#). Morover, if Ω is such a component
and λ G Ω, then the homotopy class containing the invertible function 0 λ ,
defined by
θ\ — λ — z for z G sp(rc),
is a generator corresponding to Ω. Since
we conclude that φ+ = 0. So T(x) =0.
D
Lemma 2.5. Let x and y be two elements in a unital C*-algebra A. If x is
invertible and if
then y is invertible and [x] = [y] in Inv(A)/Invo(A).
Corollary 2.6. If λ is in the unbounded component of C \ sp(#), then
λ — x G Invo(A).
Lemma 2.7. For any d > 0, any C*-algebra A and normal element x G A
and T(x) = 0 , ifyEA such that
\\x-y\\ < d,
448
HUAXIN LIN
then for any bounded component Ω ofC\sp(x) and λ G Ω with dis(λ, sp(a )) >
d,λ~j/G Invo(A), moreover, the unitary part of the polar decomposition of
λ-y is in U0(A).
Proof Since x is normal, it is known that
1
||(λ - a;)" 1| < l/dis(λ,sp(z)) < 1/rf.
If \\x - y\\ < d, then ||(λ - x) - (λ - y)\\ < d. By 2.5, λ - y is invertible and
[λ - y] = [λ - x] = 0
in Inv(A)/ Invo(A).
Therefore, λ - j / G Invo(A).
D
Lemma 2.8. For any e > 0, b > 0, an analytic function f on a region X
and a continuous function g on X, there exists δ > 0,
(1) i/ # and y are two normal elements in a (unitaΐ) C*-algebra A with
sp(rr), sp(y) C X and \\x\\, \\y\\ < b such that \\x — y\\ < ί, then
\\g(x)-g(y)\\<e;
(2) if p G A is a projection and \\px — xp\\ < J, then
\\f(pxp)-pf(x)p\\<e.
(In the above, note that if / = λ is a constant, f(pxp) = Xp )
Proof. (1) is the same as [Lin6, 1].
(2) If δ is small enough, by [Lin6, 4], there is closed curve Γ G X such
that
dist(Γ, sp(x) U sp(pxp)) > d > 0,
pf(x)p = 1/27Γ* I f(λ)p(λ - x)-ιpd\
and
fipxp) = l/2πi
If λ G Γ, by [Lin6, 4], again,
= \\\p(X-x)-1p-(Xp-pxp)-1](X-x)(X-x)-1\\
< ||p(λ - *ΓMp(λ -x)-(X-
x)p)(X - z Γ i
+ ||p(λ - xy'pKX - x)p - (Xp - pxpY'piX - x)p)(X - x)'1 \\
+
\\(Xp-pxp)-1[(X-x)p-p(X-x)}(X-x)-1\\
< δ/d2 + δ/(d - δ)d.
APPROXIMATION BY NORMAL ELEMENTS
449
Therefore
\\f(p*p)-pf(x)p\\ < 11/11 (length(Γ)).(l/2π)(l/rf2 + l / ( r f - ί ) φ .
D
Remark 2.9. Lemma 2.8 (1) remains true if g is analytic, x and y are not
assumed to be normal, but then δ may depend on x.
2.10. Let P be a polynomial of z and z*. For any element x in a C*-algebra
-A, we define P(x) to be the corresponding linear combination of xn(xm)*.
Notice that we do not assume that xx* — x*x. In fact, we should view P as
a linear combination of zn(z*)m (note that zn appears before (z*)m). Similar
to [Ch, 2], one can show that for any e > 0 and b > 0, there is a δ > 0 such
that
\\P(x) - P(y)\\ < e
whenever ||α; — y\\ < δ, \\x\\ < b and \\y\\ < b.
Lemma 2.11. Let f be a continuous function in C(D) for some disk with
the center at the origin and radius r > 0. For any e > 0, there exist δ > 0 and
a polynomial P of z and z* such that for any C*-algebra A and a normal
element x £ A, if \\x\\ < r,p is a projection in A with the property that
sp(pxp) C D and
\\px - xp\\ < 5,
then
\\pf(x)p-P(pxp)\\<e.
Proof Prom the Stone-Weierstrass theorem, there is a polynomial P of z
and z* such that
\\f-P\\D<φ.
It is routine and standard that if δ is small enough,
\\pP(x)p-P(pxp)\\<e/2.
Therefore
\\pf(χ)P - P(pχp)\\ < \\pf(χ)p -PP(X)P\\
+ \\pP(x)p - P(pχp)\\ < e.
D
450
HUAXIN LIN
3
Approximation in A ® K.
The main result in this section is Theorem 3.13. We first consider the case
that the spectrum of x is a square. In this case, y and z are constructed
by "cutting" the spectrum of x into small pieces (3.1, 3.3, 3.4). Then,
by using conforming mappings, we can deal with the case that sp(rr) is
homeomorphic to a square (3.5). For the case that sp(a ) is an annulus, the
method is somewhat different (3.6, 3.7, 3.8 and 3.9). Then we deal with
the case that sp(rc) is a square with finitely many holes. This is done by
"cutting" the spectrum into several pieces each of which is homeomorphic
to an annulus. The general idea of "cutting" spectrum comes from [BD].
However, techniques used in this section come from other sources such as
[BDF], [GL] and [Ph2].
Some of the statements in this section are somewhat complicated (two
C*-algebras A and B appear). It would be simpler if B = A or p = 1.
The following lemma looks similar to Lemma 5.2 in [BD]. But 3.1 and its
proof are inspired by [BDF, 7.4].
Lemma 3.1. For any e > 0 and η > 0, there is δ > 0 such that for any
unital C*-algebra A and x G A with \\x\\ < 1, if
C [-6,c], sp(Λ2) C [-α,/3],
where hλ = Re(x) and h2 = Im(n ), and byc,a and β are positive, and
\\xx* - x*x\\ < 5,
then there is a projection q E M2(A) such that
\\q(x Θ ih2) - (x Θ ih2)q\\ < e
and
sp(Re[q(x Θ ih2)q}) C [-77, c + 77],
sp(Im[q(x ® ih2)q\) C [-α,/?],
sp(Re[(l-q)(x®ih2)(l-q)])c[-b-η,η]
and
sp(Im[(l - q)(x φ ih2)(l - q)]) C [-a,β\.
Furthermore, if x is normal and we denote by Pi the spectral projection of
x®ih2 (in A**) corresponding to the subset
APPROXIMATION BY NORMAL ELEMENTS
451
and by P2 the spectral projection of x Θ ih2 (in A**) corresponding to the
subset
ifc = {ξ:Re(ξ)€ [-6-17,17]},
then Pι>q
and P2 > (1 - q)
Proof. Suppose that
\\xx* - x*x\\ < δι
where δι is a positive number to be determined. Let δ2 be another positive
number. Set
(
1
if δ2 < t < oo
0
ifί<-ί 2
linear iί —δ2 <t <δ2
If δ3 > 0 is given, we can choose δλ small enough such that
Wf(hi)h2 - Λ2/(Λi)|| < is,
We denote a = /(Λi) and
/
and Wf^x
α
- x/(/n)|| < δ3.
(a(l-)
q
1-α J
-[(a(l-a)y"
We have
q(x Θ iΛ2) — (x θ &7i2)?
(
^11 # 1 2 J
#21 #22 /
where
xn
1/2
12
=ax- xa, x12 = (o(l - α)) i/ι 2 - iΛ 2 (o(l - a)) '
a?21 = (α(l - a))^Hh2
- ih2(a(l
1 2
- /i x (a(l - α ) )
1
2
- α ) ) ' + (α(l - α ) ) / ^
and
#22 = (1 — a)ih2 — ih2(l — α).
If ί 2 and δ3 are small enough, by a direct computation, we obtain
||ςί(rc ® ih2) - (x ® i/i 2 )g|| < e.
Let
0
ahMl-
1/2
,
452
HUAXIN LIN
It is easy to see that
| α M β ( l - α))
1/2
|| < *2
and
1
ι2
|(α(l - ά)) '^^!
- ά)) ' \
< δ2
Therefore ||ω|| < 2δ2. Notice that
Re(q(x 0 zh2)q) = ^
^
_ α))1/2
_a))1/afci(fl(1 .
(β(1
So
Re(g(rr θ ^2)9) — (a/iia θ 0) = ω.
Since a/iχ = h\a and /ii < c, we have
(x θ ih2)q) <c + 2δ2.
On the other hand, if r < — J 2 , then
IKr-αΛiαθOΓ^l/lr + ίal
and
(11 - [r - (ahxa θ O ) ] " 1 ^ - Re(q(x ® ih2)q))\\
< II[r - αΛiα θ O ] " 1 ^ < ( l / | r + ί 2 |)
2ί 2
For 77 > 0, if δ2 is small enough, whenever
r < -!/(< - ί 2 / 2 ) ,
r — Re(<7(# 0 i/ι2)^) is invertible. Therefore
s p ( R e [ φ φ ih2)q}) C [-77, c + η).
Similarly,
sp(Re[(l - q)(x φ ih2)(l
- q)]) C [-6 - 17, r,].
Moreover, since Im[g(α; φ 2/12)9] = 9(^2 θ h2)q,
sp(Im[q(x 0 2/12)^]) C [—α, /?].
Similarly,
( I [ ( l ) ( Λ ) ( l ) ] )
C [-«,yS]
a))1/2
J
APPROXIMATION BY NORMAL ELEMENTS
453
Finally, if x is normal, and if δ2 < η, one sees easily that
and
Pι>q
P2>(l-q).
D
In Lemma 3.1, we also have
\\x Θ ih2 - [q(x Φ ih2)q + (1 - q){x Θ ih2)(l
- q)]\\ < e/2.
Lemma 3.1 allows us to "cut" the spectrum of x into two vertical strips. But
in order to do that, we have to add to x a normal element with spectrum
contained in a vertical line. In the following corollary, by repeating this, we
can "cut" the spectrum of x into k vertical strips. But for each cut, we have
to add a normal element with spectrum contained in a vertical line. In 3.2,
we only do this to a normal element.
Corollary 3.2.
Let
X = {X: | R e λ | < l , | I m λ | < 6},
where b > 0. For any e > 0 and η > 0, there is δ > 0, for any unital
C*-algebra B and a normal element x E B with
||Re(a;)||<l
and
||Im(x)|| < 6,
if
\\px - xp\\ < δ,
and if
- 1 = t 0 < ίi < ί2 < -. < h = 1,
then there are mutually orthogonal normal elements yi,y2, ...,yk-i
€
M2k-ι_ι(pBp),
mutually orthogonal projections pι,p2, ...,pk E M2k-i(pBp)
and normal elements zx,z2,...,zk
E M2k-i(B) with sp( ^ ) C X satisfying
C {tά +ia : \a\ < b}J = 1,2,..., k-
1,
such that
\\pjZj -ZjPjW
< e
and
\\x Θ yi Θ
Θ yk-i ~ Pi*iPi θ P2Z2P2 θ
® pkzkPk II <
454
HUAXIN LIN
where
Xη = {\:dist(\,X)
<η}
and
Rj = {λ : tj_r -η/2<Reλ<
tj + ry/2, | Im λ| < b + η/2}.
Furthermore,
(1) if we denote by Pj the spectral projection of Zj (in B**) corresponding
to Xη Π i£p then Pj > Pj.
(2) if A is a C* -subalgebra of B, p G A, pf(x), f(x)p G A for any
f G C(sp(x))1 then we can have y{ G M2k-i_1(pAp)
and pjg(zj), g(zj)p G
M2k-ι(pAp) for any g G C(sp(zj)).
Proof The proof is a repeated application of Lemma 3.1. Notice that, for
any element z G A and any projection q G A, Im(qzq) = q(Imz)q. We also
notice that it follows from [Lin6, Lemma 4] that if δ is small enough then
sp(q(x®ih2)q) C Xη and sp((l—q)(x@ih2){l—q)) C Xη. The normal element
Zj is a direct sum of x with a normal element with spectrum contained in
finitely many vertical line segments.
D
Now we will "cut" the spectrum of x vertically as well as horizontally. So
the spectrum of x is cut into small pieces. Note that for each cut, we have
to add a normal element with spectrum contained in a line segment. Note
also that 3.2 is not used in the proof of 3.3.
L e m m a 3.3. Let
X = {z: \Rez\ < l / 2 , | I m * | < 1/2}.
For any e > 0, there exist δ > 0 and an integer fc, for any unital C*-algebra
A and an element x G A, if
| | R e ( a O | | < l / 2 , ||Im(x)|| < 1/2
and
\\xx* — x*x\\ < 5,
there are mutually orthogonal normal elements yi,2/2? •• 52/m £ Mk{A) with
sp(j/i) contained in a straight line segment in X and a normal element z G
Mk+ι(A) with finite spectrum sp(z) C X such that
||sθϊ/iΘy2θ
- Θ y m - 2 | | < c.
Proof We will apply Lemma 3.1 repeatedly. Let eλ and 771 be two positive
numbers. For e — eλ and η — 771, let δ — δγ > 0 be the number in Lemma
3.1. We will use the notation in Lemma 3.1. Set
zγ — i[q(x Θ ih2)q - 1/4],
and z2 = i[(l - q)(x θ ih2)(l
— q) + 1/4].
APPROXIMATION BY NORMAL ELEMENTS
455
We have
\\zjz*j-z*jzj\\<2e1+δu
||Re^ ||<l/2
and | | I m ^ || < 1/4 + ι/χ,
j = 1,2. Let 62 and η2 be positive numbers. If €1,77! and δλ are small enough,
by applying Lemma 3.1 again, there are projections 9i,92?PijP2 such that
q\ < q, q2 < (1 - q), Pi=q~qi and p2 = (1 - ςr) - #2, and
θilmz^qjW
Θ ilmzά)qά)\
^ )^)]
)^)]
Θ i l m ^ )^)]
< e2,
C [-%, 1/2 + τ?2],
C [-1/2 - r?2,r?2],
C [-1/4 - τ/2,1/4 + τ?2],
C [-1/4 - r/2,1/4 + τ?2],
j = 1,2. Denote Xj = (l/i)qj(zj ®ilmzj)qj + (—l) 1+J l/4, j = 1,2 and
1+
Zj)Pj + (-l) U/4, j = 1,2. Then
4
x®ih2@iIm[(l/i)z1
Xk
+ 1/4] Θ iIm[(l/z)^2 - 1/4] ~Σ
2e
2e
< i+ 2
k=l
1 2
Notice that there are complex number aά = (l/2)(l/2) / e^J7Γ/4^ such that
\\XJ - α i 9 i | | < 2(1/4 + η2)
and
: 2(1/4+ r/2).
By continuing to apply 3.1, one sees that for any e > 0, there is an integer
k and δ > 0, if
||£*z-££*H < δ,
there are mutually orthogonal normal elements yi, ?/2, ...,ym G Mk(A) with
sp(yi) contained in a straight line segment in X and mutually orthogonal
projections ex, e2,..., efc+i E Mfc+i(A) and complex numbers λ x , λ2,..., λfe+i G
such that
M-l
ym
-
e.
D
Corollary 3.4.
= {λ : |Reλ| < l/2,|Imλ| < 1/2}.
456
HUAXIN LIN
For any e > 0, there exist δ > 0 and an integer A;, for any unital C*-algebra
A of real rank zero and an element x G A with
\\Rex\\ < 1/2 and ||Im*|| < 1/2,
if
\\x*x-xx*\\ < δ,
there are normal element y G Mk(A) and normal element z G Mk+i(A) with
finite spectra sp(y),sp(z) C X and
\\x®y-z\\ < e .
Proof. Since A has real rank zero and a normal element with spectrum contained in a line segment has the form aa+β, where a is a selfadjoint element
in A and a, β are complex numbers, the y^s in Lemma 3.3 are approximated
(in norm) by normal elements in Mk(A) with finite spectra.
D
Lemma 3.5. Let X be a compact subset of the plane which is homeomorphic
to the unit disk. For any e > 0, there exist δ > 0 and an integer fc, if
x i$ a normal element with sp(a ) C X in a C*-algebra B and if p is a
nonzero projection in a C*-subalgebra A of B with real rank zero, such that
pf(x), f(x)p G A for any f G C(sp(z)),
\\px — xp\\ < δ and sp(pxp) C X,
then there are normal elements y G Mk(pAp) and z G Mk+\{pAp) with finite
spectra sp(y),sp(^) C X such that
\\x®y-z\\
<e.
Proof. Set
S = {λ : I Re λ| < 1/2, | Im λ| < 1/2}.
For any e > 0, let Xe be a region such that
X C Xe, Xe C {ξ : dist(£,X) < e/2}
and there is a conformal mapping / from Xe onto S. There is η > 0 such
that
f(X) C {ξ : |Re(f)| < 1/2 - 17, |Im(O < 1/2 - f|}
Choose δ > 0 such that
\\pf(x)p-f(pxp)\\<η/4:
APPROXIMATION BY NORMAL ELEMENTS
457
(see 2.8). Since Re(pf(x)p) = pRe(f(x))p, we may also assume that
\\Re(f(pxp))-pRe(f(x))p\\<η/A.
Since x is normal, we have
\\Re(f(pxp))\\ < \\PRe(f(x))p\\+ηβ < \\Re(f(x))\\+η < \\Re(f)\\x + η.
Similarly, we have
By 3.4, for any σ > 0, if δ is small enough, there exist an integer k (which does
not depend on A, B or x but does depend on X and e), and normal elements
Hi £ Mk{pAp) and z\ G Mk+ι(pAp) with finite spectra sp(yι), sp(zχ) C S
such that
\\f(pxp)®Vi-z1\\ <σ.
By 2.8, if σ is small enough,
Since sp(f~1(y1)) sp(f~1(z1)) G Xe, by changing the spectrum of /"^(yi)
and f~ι(zλ)
slightly (within e/2), there are normal elements y and z as
required.
D
The following lemma is inspired by [Ph2].
Lemma 3.6. For any e > 0 there exist δ > 0 and d > 0, for any unital
C*-algebra A and x G A with the polar decomposition x = uh such that
0 < a < h < 1 (so u is a unitary), if
\\uh - hu\\ < δ,
then exists y G M2(A) with
sp(y) C {reiθ : a < r < 1, - π + d/2 < θ < π - d/2}
such that
\\x@x* — 2/|| < e.
Proof Let
(
. _ ίu θ\ ( cosα sinαΛ /?/* θ\ (cosa— sinα\
l o i i l — sin a cos α i I 0 1 y I sin α cos a J '
458
HUAXIN LIN
If |α| < τr/2, by [Ph2, 5], - 1 0 sp(u(a)). Let θ be the largest number in
[0, π] such that eiθ G sp(u(a)) and let d = π — θ. It is easy to see that
\\u(a)(h ®h) -x® x*
if a is close to π/2.
One can verify that
||ti(α)(Λ θ h) - (h θ Λ)tι(α)|| < 2<5.
Let λ = reiβ, where /? £ [-π + d/2, π - d/2], then
[λ - tx(α)(Λ θ ft)]*[λ - tx(α)(Λ θ h)]
= r2 + (h® h)2 - reiβ(h θ Λ)tx(α)* - re""i/3ti(α)Λ(θΛ)
> r 2 + (/ιθ Λ)2 - r(Λ θ h)1/2(ei(3u(a)* + e-iβu(a))(h
θ /ι) 1 / 2 - 2rδ1
>r2 + (h® h)2 - 2r(h φ h) cos(d/2) - 2rδu
> 2r(h θ Λ)[l - cos(d/2)] + [r - (h θ Λ)]2 - 2rJx
> 2rα[l - cos(d/2)] - 2 r ί i ,
where δx = ||(/ιθ/ι) 1/2 ϊx(α) -n(α)(/ι0/ι) 1 / 2 ||. Therefore if 5 is small enough,
[λ - tx(α)*(Λ θ Λ)][λ - ti(α)(Λ θ Λ)]
is invertible. Similarly,
[λ - u(a)(h e Λ)][λ - tx(α)(Λ θ Λ)]*
is invertible. Hence λ 0 sp((α)(/i θ Λ)). It is clear that if |λ| > 1, then
λ 0 sp(u(a)(h θ h)). Note U(Q ) is a unitary. Since h 0 fo > α, if |λ| < α,
(α)(ΛθΛ)).
D
Corollary 3.7. Let x be a normal element in a {unitaΐ) C*-algebra B with
polar decomposition x — uh such that 0 < a < h < b for some positive
numbers a and b (so u is a unitary). For any e > 0, there exist d > 0 and a
normal element y G M2(B) with
sp(y) C {reiθ : a < r < 6, - π + d/2 < θ < π - d/2}
such that
\\x®x*-y\\
<e.
Furthermore, for any η > 0, there exists δ > 0 such that if there is a nonzero
projection p G A with
\\px - xp\\ < ί, pf(x), f(x)p G A,
APPROXIMATION BY NORMAL ELEMENTS
459
where A is a C* -subalgebra of B containing the identity of B, then
II(pθp)y-y(pθp)|| < η
and (p@p)g(y), g(y)(p®p)
E M2(A)
for any g E C(sp(y)).
Proof. Note that in Lemma 3.6, if α; is normal, u(a) commutes with h®h for
every a and every u(a) is untary. So the element y = u{a)(hθ h) is normal.
Note that h = (x^x)1^2 and u = α /i" 1 . So h — fλ(x),u = f2(x) for some
/ij/2 ^ ^(spί^))- Hence, pu,up,ph,hp £ A. A direct computation shows
that (pθp)/(y),/(ί/)(pθp) G M2(A) for any / E C7(sp(y)). Furthermore,
for any 77, since a; is normal, there is δ > 0 such that if
\\px - xp\\ < ί,
then
||pιt - txp|| < η/2
and
\\ph - hp\\ < η/2.
So
II (p θ p)tx(α) - tx(α) (p θ p) II <
and
II(p θp)(Λ θ Λ ) - ( f t θ h)(p®p)\\ < η/2.
We may take
y = u(a)(h(Bh)
for some a.
D
Lemma 3.8.
X = {re^ : 0 < α < r < 1, - π < θ < π}.
JFbr an?/ e > 0, ίΛere ea isί ί > 0 and an integer k, for any unital C*-algebra
β, any C*-subalgebra A of B with real rank zero containing the identity of
B, and a normal element x E B with the polar decomposition x — uh such
that 0 < a < h < \\x\\ (so u is a unitary), if p E A is a projection such that
\\px — xp\\ < δ, sp(pxp) C X and pf(x) f(x)p G A
for any f E C(sp(a;)), then there are normal elements y E Mk(pAp) and
z E Mk+2(pAp) with finite spectra contained in X such that
\\pxp θ px*p ®y - z\\ <e.
460
HUAXIN LIN
Proof. For any e > 0 and η > 0, by 3.7, if δ is small enough, there exist d > 0
and a normal element j/i G M2(B) with
sp(yi) C {reiθ : a < r < 6, - π + d/2 < θ < π + d/2}
such that
||arΘs*-ϊ/i|| <e/3, ||(pθp)yi - yi(p@p)\\ <η
, f{yi)(p®p) G
Set g = p θ p and set
Y = {re^ : α - e/3 < r < b + e/3 and - π + rf/4 < β < π - d/4}.
Then, if 77 is small enough (so δ has to be small enough),
C
r.
Notice that F is homeomorphic to the unit disk. By Lemma 3.7 and Lemma
3.5, if η is small enough, there are normal elements y' E Mk(qAq) and
z' G Mk+1(qAq) with finite spectra sp(y),sp(z) C Y such that
for some positive integer k. Therefore
\\pxp®pχ p φ y' - z'\\ < 2e/3.
By changing the spectrum of y' and z' slightly (within e/3), we have
\\pxp@px*p®y-z\\ <e,
where sp(y),sp(z) C X, y G Mk(pAp) and z G Mk+1(pAp).
D
The idea of using a path of unitaries in the following proof is taken from
[Ph2].
Lemma 3.9. For any e > 0, there exist δ > 0 and an integer k, for any
unital C*-algebra JB, any C*-subalgebra A of B with real rank zero containing
the identity of B and a normal element x G B with the polar decomposition
(in B) x = u/i, where 0 < α < h < 1 (α and 6 are positive numbers) and u
is a unitary, and if p is a projection in A such that
\\px — xp\\ < δ, pxp G Invo(pAp) and pf(x), f(x)p G A
APPROXIMATION BY NORMAL ELEMENTS
461
for any f G C(sp(α;)), and
iθ
sp(pxp) C X = {re
: α < r < 1, - π < (9 < π},
ίften £ftere are normal elements y G Mfc (pAp) and z G Mfc+χ (pAp) with finite
spectra contained in X such that
\\pxp Θ y - z\\ < e.
Proof. Let e be a positive number. For e/16, let δ and k be the numbers in
Lemma 3.6. Notice that δ and k does not depend on x or the C*-algebra A.
We may assume that δ < e/8. Set
\z\ < l + e/4}.
Y = {z:a-e/4<
Let a; = u/i be the polar decomposition. If
then
||pί/ — ί/p||
and
||p/i — hp\\
are small. We assume that
< e/4,
- t)h +1] - tx[(l - i)Λ + t]p|| < δ.
Furthermore, by Lemma 3 in [Linβ], we may assume that sp(pxtp) C Y
(notice that (1 - ί)α + 1 < (1 - t)h + ί), where 0 < t < 1, xt = u[(l - t)h +1].
Notice that δ0 can be chosen such that it does not depend on x or the
C*-algebra A (but depends on e, a and b). Notice also that, if δ0 is small
enough, v = pup\pup\~ι G Uo(pAp). Suppose that {v(t), 0 < t < 1} is a
path of unitaries in pAp such that i?(0) = υ and τ (l) = 1. Set
_ fu[(l-t)h
~\v(t-1)
(f)
X[)
+ t] if 0 < t < 1
if 1 < * < 2 "
So α (O) = x and x(2) = 1. For any e > 0, let x 0 = £ and Xi = a (ti), i =
1,2,..., m + 1 such that 0 < U < 1, i = 1,2,..., n, 1 < U < 2, i = n + 1, n +
2, ...,m, ί m + i = 1,
||Xi - Xi-i\\ < e/4, i = l,2,...,n
and ΐ = n + 2, ...,ra + 1
and
\\pxnp - pxn+1p\\ < e/4.
462
HUAXIN LIN
From pf(u)J(u)p,pg(h),g(h)p
E A for any / E sp(sp(u)) and# E C(sp(Λ)),
it is easy to see that pf(xi),f(xi)p E A for any / E C(sp(#i)). Notice that
from [Lin5], ceτ(A) < 1 + e. Therefore, the parth {υ(t)} can be chosen such
that the integer m depends on e only. Set
Vi = Σ ®£iP*iPΘ Σ ©SiP^iP
By 3.8, there are normal elements y2 £ ^fcm(pAp)
with finite spectra contained in Y such that
an
d
Let
2/o = pa?p θ j / i θ p ,
and i/ό = Σ © S ^ i P © Σ ©SoPsJ P
Then
llz/o — i/όll < e/4By 3.8 again, there are normal elements y3 E Mk(m+i)(pAp) and z2 E
-ft^(M-i)(m+i)(p^p) with finite spectra contained in Y such that
Ilϊ/£θy3-*2|| <e/4.
Then it is easy to see that there is a unitary U E M2+(k+i)(m+2)(pAp) such
that
Hi/ θ p θ zλ θ 2/3 - U*(y2 θ z2)C/|| < 3e/4.
By changing the spectrum of y2)2/3?^i a n d z2 slightly (within e/4), we may
assume that sp(2/2),sp(y3),sp(^i),sp(^2) C X.
D
3.10. For the convenience, we will consider a special compact subset of the
plane which is a square with finitely many holes. Let
X' = {z:0<Rez<
l,\Ίmz\ <b}\ Ukjz=1{z : \{2j - I)/2k - z\ < r},
0 < r < min(6,1/4Λ), and X = {z -1/2 : z e X1}. So X is a square with k
holes. Let d = [fe2 + (l/2Jt)2]1/2 < 3/4(l/fc - 2r) and
r = Ukj=1{z : r/2 < |(2j - l)/2k - z\ < 4/3d}.
Define Y = {z — 1/2 : z E y } . Y is a union of finitely many annuli but holes
are disjoint. There is a retraction / : Y -» X.
Lemma 3.11. Let Ω be a compact subset of the plane which is homeomorphic to the subset X described in 3.10. For any e > 0, there exist δ > 0 and
APPROXIMATION BY NORMAL ELEMENTS
463
an integer L with the property that, for any normal element x in a unital
C*-algebra i?, if sp(x) C Ω and p is a projection in a C*-subalgebra A of B
with real rank zero containing the identity of B such that
\\px - xp\\ < δ, p/(x), f{x)p G A
for any f G C(sp(x)), and
λp — pxp G ϊnvo(pAp)
for λ 0 X, then there are normal elements y G Mι(pAp) and z G Mι+ι(pAp)
with finite spectrum sp(y),sp(z) C Ω such that
\\pxp®y-z\\ <e.
Proof We first claim that it suffices to show the case that Ω — X.
There is a homeomorphism / : Ω -> X. For any σ > 0, if δ is small enough,
Suppose that 3.11 holds for Ω = X. Then, for any η > 0, if σ is small enough,
we have y and z as in the conclusion of the lemma but with inequality
\\pf(x)p®y-z\\ <η.
There exists a continuous function / : D —> C such that I? is a disk with
the center at the origin containing X and f\x = f~λ. By 2.11, there is a
polynomial P (or rather a linear combination of zn{z*)m) of z and 2*, such
that
\\pf(f(x))p-P(pf(x)p)\\<e/4,
if σ is small enough. We may further assume that
Hence
i/(y)-P(y)||<e/4
and | | / » - P ( z )
< e/4.
Note also that f(f(x)) — x- Thus, by 2.10, if σ is small enough, we have
Therefore
\\pχpΦf(y)-f(z)\\
< \\P(pxp) ® P(y) - P{z)\\ + e/A + e/4 + e/4 < e.
464
HUAXIN LIN
Note sp ί f(y)j , sp (f(z)) C Ω and /, / and P depend only on Ω and X.
This completes the proof of the claim.
For the rest of the proof, we assume that X and Y are as in 3.10.
Set
ti = - 1 / 2 + i/fc, i = 0,1,2, ...,fc.
For any r/4 > η > 0, applying 3.2 (if \\px — xp\\ < ί), we obtain normal
elements yi,y2> ?yfc-i £ M2fc-i_i(pAp) with
s
P(Vj) C {i, + is : -6 < s < &},
mutually orthognal projections Pi,P2,'~iPk £ M2fc-i(/λ4jp) and normal elements Xι,x2,...,Xk ε M2fc-i(J5) such that Pjf(xj),f(xj)Pj
€ M2fc-i(^4) for
any / G C ί s p ^ )),
sp^
) C X,
\\pjXj - XjPj\\ <
η/4
C {λ : t^i -η/4<Reλ< tά + η/4}
and
\\pχp θ 2/i θ
θ y*_i -Pi^iPx θ
ΘPkXkPkW < η/16,
where
and
Rj = {λ : t^.i - τ?/4 < Reλ < tj + η/4, -6 - η/4 < Imλ < 6 + rj/4}.
Since A has real rank zero, we may assume that each yj has finite spectrum.
Furthermore, by Lemma 3 in [Lin6], we also have
\\(xPj -PjXjPjΓ^l < i/dtat(λ, j g .
If δ < 7?/16, from Lemma 3 in [Lin6] and 2.7,
Xq - pixipi ®p2x2P2 θ • ®Pk%kPx e Inv o (M 2 ^i (pAp)),
where λ 0 Xη and q is the identity of M2k-i{pAp). Let
*i = Pi^P, - [(2i - l)/2* - 1 / % ,
Then, if 7? is small enough,
\\ZJ\\
< (4/3)d. Set
Yj = {λ : r/2 < |λ - [(2j - l)/2fc - 1/2]| < (4/3)d}
APPROXIMATION BY NORMAL ELEMENTS
465
By 2.6, if i φ j and
λ G {λ : |λ - (2j - I)/2k - 1/2| < (4/3)d},
then
λPi -PiXiPi € lnvo(piM2k-i(pAp)pi).
Since
Xq - pxxipi Θ
ΘPkXkPx € Invo(M2fc-i
by 2.3,
ifλ&Yj. Set
y 0 = {λ : r/2 < |λ| < (4/3)d}.
Then
λPj - Zj e lnvo(M2k-i(pAp)),
Ίΐλ $Y0. Denote by Pj the spectral projection of x'j = x7 — [(2j — l)/2k —1/2]
(in B**) corresponding to the subset Yo. By 3.2, we have Pj > pj. So PjXj is a
normal element in B** with s p ( P ^ ) C lo and Zj = PjPjx'jPj. Furthermore,
Pi/ίP,-*;) = PiPi/ίxJ) - P , / ( ^ ) € M ^ - ! ^ ) , / ^ ^ e Af2*-i(A) for all
continuous functions / G C(yί)). We are now ready to apply Lemma 3.9. If
δ is small enough, there are normal elements y'j G Mx,(pjM2fc-i(-A)pj) and
normal elements y" G M L + I ^ M ^ - I ^ ) ^ ) with finite spectra contained in
Yj such that
\\PjXjPj ® Vj - i/j\\ <V/^
for some integer L {pjXjPj — zά + [(2j - ί)/2k - l/2]pj). This implies that
\\pxp Θ Σ θj-ίyi θ Σ ®i=i»i ~ Σ ®i=i»i I < */2
Finally, set
and
Clearly, sp(y),sp(^), X C F. There is also an integer L (which depends only
on e) such that y G ML(pAp) and z G ML+ι(pAp). Moreover,
\\pxp®y-z\\ <η.
466
HUAXIN LIN
There is a retraction R : Y -» X. Let R' : D -> C be a continuous function
such that D is a disk with center at the origin containing Y and R'\γ = R
There is a polynomial P of £, t* (or rather a linear combination of £ n (£*) m ,
see 2.10) such that
| | P - R'\\D < e/4.
By 2.10, if η is small enough,
\\P(pxp)®P(y)-P(z)\\<e/4.
By 2.11, if δ is small enough,
\\P(pxp)-pR'(x)p\\<e/4.
Note i?'(x) = x. Therefore we have
\\pxp®R'(y)-Rf(z)\\<e.
Since R'\γ = # , sp(Λ'(y)), sp(Λ'(s)) C X
D
Lemma 3.12. Let Ω be a compact subset of the plane. For any e > 0, there
exist δ > 0 and an integer L such that for any (unitaί) C*-algebra A of real
rank zero and a normal element x in a C*-algebra B D A with sp(a ) C Ω, if
p G A is a projection and if pf(x), f(x)p G A for any f G C(sp(a;)),
\\px — xp\\ < δ and λp — pxp G lnvo(pAp)
for λ ^ Ω, then there are normal elements y G ML(pAp)
with finite spectrum sp(y),sp(z) C Ω such that
WpxpΘy-
and z G ML+ι{pAp)
z\\ < e.
Proof. For any e > 0, there are finitely many closed balls JBi, B2,..., Bn with
centers in Ω and diameters less than e/2 such that
Ω C UjLxBi.
We may write that U™=1ί?i = \Jψ=1Xj, where each Xj is a connected component of U™=1I?i. So we may write x — Σ j = i xj > where each xk is normal
and sp(α j) (in a corner of B) is a subset of Xj. Since each Xj is a union of
some B^s, Xj is homeomorphic to the subset X described in 3.10. We may
assume that
APPROXIMATION BY NORMAL ELEMENTS
467
by taking smaller δ. Let pi,p 2 , ~">Pm be m mutually orthogonal projections
such that
m
Σp3
=1
and pάxά = Xjpό = s,, j = 1,2,..., m.
i=i
For any 7/ > 0, by taking a small J, we may assume that
Note there is g^ G C(sp(x)) such that pj = gj(x). Thus PjPPj £ AΛfη is small
enough, there is fj G C(sp(pjppj)) such that qj = fjiPjPPj) is a projection
in A (cf. [Eff, A8]). Furthermore, qjPjPPj = PjPPjqj is invertible in qjAqj.
Since pjppjf(x)J{x)pjppj
G ^ for any / G C(sp(x)), ?,•/(*), /(ar)(fc G A.
T h e n
E
Let <? = ΣΓ=i 9i
«/(*)> / ( ^ ί ^
For any 1 > σ > 0, if η is small enough,
\\q-p\\<σ.
There is a unitary w e A such that
||w — 1|| < 2σ
and t ί ; * ^ = p.
We claim that it suffices to show the case that q = p.
Suppose that we have
where y G ML(gAg) and 2? G Mx.^^gAg). Let W\ = diag(ίi;, w, - , w) (there
are L copies of w) and W2 = diag(w, w
,w,w) (there are L + 1 copies of
w). We have
\\w*(qxq)w 0 WfyWx - W2ZW2W < σ
and
||tί;*gxg^ - pxp\\ < 2σ.
Thus
W*yW1 - W;zW2\\ < 3σ.
Note that W*yWλ G ML{pAp), W*zW2 G M L+ i(pΛp), and 8p{WfyWi) =
sp(y), sp(PF2*2:W2) = sp(z). This proves the claim.
So, without loss of generality, we may assume that p = Σj=i 9
9j < Pj. Next we show that if dist(λ, Ω) > 25,
G lnvo(qjAqj).
468
HUAXIN LIN
If λ is in the unbounded component of C \ sp(x J ), then Xqj
lnvo(pjApj). Set
X'j = {ξ: distfc Xj) < ί/2}, i - 1, 2, ...m.
If λ is in the bounded component of C \ Xj, then λ is in the unbounded
component of C \ sp(xk) for k φ j . So λpk — pkxpk € Invo(pkApk) for A: 7^
j . It follows from 2.3 that Xpj — PjXPj G Invo(pjApj). We note that Xj
is homeomorphic to the subset X in 3.10. We then apply 3.11 to each
Xj.
D
Theorem 3.13. Let A be a C*-algebra of real rank zero and let x be a
normal element in A. If T(x) — 0, then for any e > 0, there is an integer
k, and there are normal elements y G Mk(A) and z G Mk+1(A) with finite
spectra contained in sp(rz ) such that
\\χ®y-
z\\ < e.
(The integer k depends only on e and sp(a ).) Moreover, the converse is also
true.
Proof. When A is unital, this follows from 3.12 immediately. Now we assume
that A is not unital. Then 0 G sp(:r). As in the proof of Theorem A in [Linβ],
for any e > 0, there is a projection e G A and a normal element x' G eAe
such that
\\x' - x\\ < 6/3.
Set
X e = {λ:dist(λ,X) < e / 3 } .
It follows from 2.7 that λe — x' G Invo(e^4e), if λ G Xe> By applying the
theorem for the unital case, we obtain normal elements x' G Mk(eAe) and
z G Mk+ι (eAe) with finite spectrum contained in Xe such that
\\x' ®y-z\\ <e/3.
Therefore
Us θ y - *|| < 2e/3.
By changing spectra of y and z slightly (within e/3), we may assume that
sp(y), sp(*) G X.
Now for the converse, suppose that there are normal elements y G Mk(A)
and z G Mk+i(A) with finite spectra contained in sp(rr). such that
APPROXIMATION BY NORMAL ELEMENTS
469
For any fixed λ E C \ sp(:r), set
d = dist(λ,sp(:r)).
Then
If e < 1/rf, by 2.7,
[λ - s Θ y] = [λ - s] e Inv(Mfc+1(Λ))/Invo(Mfc+1(A)).
Since λ — z has finite spectrum, λ - z G Invo(Mjfe+1(A)). This implies that
\-x@y
G
Since λ — y has finite spectrum, λ - j / E I n v o ( M f c ( A ) ) . It follows from 2.3
t h a t λ - x e Invo(A). So by 2.4, Γ(x) = 0.
D
4. C*-Algebras with (FN) and Weak (FN).
We start with the following definition.
Definition 4.1. (cf [BU]). We say a C*-algebra A has the property (FN)
if every normal element can be approximated by normal elements with finite
spectra.
It is clear that all W*-algebras and AVF*-algebras have the property (FN).
Some examples of separable C*-algebras which have the property (FN) can
be found in [Phi]. It is also clear that every commutative AF-algebra
have the property (FN). But it is not known that all AF-algebras have the
property (FN) (see [Linβ]). From 3.13 we see that if U(A)/U0(A) Φ 0, then
A can not have (FN), even though A has real rank zero. This, of course,
is also clear from [Phi], since not every C*-algebra with real rank zero has
(FU). As we stated in Section 1, related to question Q, we give the following
definition:
Definition 4.2. A simple C*-algebra A is said to have the property weak
(FN) if every normal element x E A can be approximated by normal elements
with finite spectra in A if and only if T(x) = 0.
This terminology is certainly borrowed from N. C. Phillips's weak (FU)
([Phi]). We are grateful to Terry Loring who pointed to us that the definition of weak (FN) for general C*-algebras will be more complicated. We
will discuss this issue elsewhere.
Recall that a simple (7*-algebra is said to be purely infinite if every projection in the algebra is infinite (one may use other definitions in [LZ] which
do not mention projections).
470
HUAXIN LIN
L e m m a 4.3. ([Lin6, Lemma 1]). Let A be a unital C*-algebra and x E A
be a normal element Then for any e > 0, if
(1) λi,λ 2 ,...,λn Gsp(α ) and |Atf — λ^l
>e,i^j;
(2) Sk is an open subset of {λ : |λ — λfc| < e} containing A*;
(3) qk is the spectral projection of x in A** corresponding to the open set S^
(4) pk is a projection in Heτ(qk);
(5) y = (1 - ΣILiP M l - ΣILiP;), then
<2e
k=l
and
<2e.
Theorem 4.4. Let A be a purely infinite simple C* -algebra and let x be a
normal element in A. Then x can be approximated by normal elements with
finite spectra if and only if Γ(x) = 0.
In other words, every purely infinite simple C* -algebra has the property
weak (FN).
Proof. By [Zh2], A has real rank zero. So, every hereditary C*-subalgebra
has real rank zero. It follows from 4.3 that for any η > 0 and δ > 0, there
are mutually orthogonal projections Pi,ί>2? •••5Pn £ A and complex numbers
λi, λ 2 ,..., λ n G sp(α ) such that
iPi + (1 - p)x(l
-p)-x <η/8,
where p = ΣΓ=i Pi»
\\px - χp\\ < δ
and for any λ G sp(rc), there is i such that
dis(λ, λ^ < η/4.
Set y = (1 — p)x{l — p). Combine 2.4 with 2.6 and 2.7, if δ is small enough,
sp(y) CXV = {\: dist(λ,sp(a;)) < η/4} and
λ(l - p) - y G Inv o ((l - p)A(l - p))
APPROXIMATION BY NORMAL ELEMENTS
471
if λ 0 Xη. By 3.12, if δ is small enough, there are normal elements yι E
Mk((l — p)A(l — p)) and y2 £ Mk+1((l — p)A(l — p)) with finite spectra
contained in Xη such that
— ϊ/all
for some integer k. By changing the spectrum of yλ and the spectrum of y2
slightly, we may assume that
and sp(yi) sp(y2) C sp(a ).
Without loss of generality, we may further assume that yι = ΣiLi \<lu
where qι are mutually orthogonal projections in (1 — p)A{\ — p) such that
Σ7=i Qi = ^ e identity of Mk({l — p)A(l — p)). Since A is purely infinite
simple C*-algebra, there is a partial isometry
v e (p Θ (1 - p) Θ ... Θ (1 - p))M f c + 1 (A)(p Θ (1 - p) θ ... θ (1 - p))
(there are A; copies of 1 — p) such that v*qtv < p{, i — 1,2, ...,n,
n
an
υ*v = Σ v*Qiv
d
w * = (1 — p) θ (1 — p) θ ... θ (1 — p)
(there are A: copies of 1 — p).
Set i4 = (1 — p) θ υ. Notice that uu* = 1 ® (1 - p)
A: copies of (1 — p)). We have
(1 — p) (there are
and ϊ/*y2^ is normal and has finite spectrum, where p\ = v*qiv. So
n
n
i=l
z=l
Therefore
χ
- Σ
Xi
u
(pi ~ p'J ~ *
< e.
D
472
HUAXIN LIN
Corollary 4.5. The Cuntz algebra On has (FN), i.e. every normal elements
in On can be approximated by normal elements in On with finite spectra.
Proof. It is known ([Cu]) that On is purely infinite simple and Kι(On) =
0.
D
One crucial result in BDF-theory is that the Calkin algebra B(H)/K has
weak (FN). The following is a generalization of this fact. The question when
corona algebras are simple is discussed in [Linl] and [Lin7],
Corollary 4.6. Let A be a σ-unital simple C* -algebra of real rank zero. If
the corona algebra M(A)/A is simple, then M(A)/A has weak (FN).
Proof. It follows from [Zh6, 1.3] that M(A)/A is a purely infinite simple
C*-algebra. So the corollary follows immediately from 4.4.
D
Definition and Remark 4.7. Let A be a separable simple unital C*algebra with real rank zero, stable rank one and with weakly unperforated
K0{A) (see [B13, 6.7.1]). Then G = Ko(A)/K0{A)tor is a simple ordered
group. By [Zh5, 1.3] and [EHS], G is a simple dimension group. Fix a
nonzero projection e G A. Let
Δ = {r e S : τ(e) = 1},
where S is the set of positive homomorphisms from G into R (see [Eff],
Chapter 4). With weak*-topology, Δ is a compact convex set. Let p G A be
another nonzero projection and let
Δ' = {r G S : τ(p) = 1}.
If Δ has countably many extreme points, then Δ' has also countably many
extreme points (see [G2, 6.17]). It is known that Δ is a Choquet simplex.
By [Al, 1.4.9], every point in Δ is a barycenter of a measure concentrated
on its extreme points. So, if Δ has countably many extreme points {τn},
then for any T E A, there is a sequence of nonnegative numbers {an} such
a
that ΣZ=i n = 1 and
n=l
Furthermore, every r E A defines a quasitrace on A (see [B13, 6.9.1]). We
will say that K0(A) has countable rank if Δ has countably many extreme
points.
The map:
θ : G -> Aff (Δ)
APPROXIMATION BY NORMAL ELEMENTS
473
determines the order on G in the sense that
G+ = {g€G:θ(g)»0}U{0})
where Aff (Δ) is the set of all (real) affine continuous functions on Δ. As in
[Eff, 4], if G φ Z, Θ(G) is order isomorphic to a dense subgroup of Aff (Δ),
provided with the relative strict order. Moreover, g < f in K0(A) if and
only if φ(g) < φ(f), where φ is the composition map:
K0(A) -> K0(A)/K0(A)tor
-> Θ(G).
By [Bl, 6.9.3], \p] < [q] in K0(A) if and only if τ(p) < τ(q) for all quasitraces
in Δ. Furthermore, for any two projections p,g 6 i , τ(p) < r(q) for all
quasitraces in Δ if and only if there is a partial isometry v G A such that
v*v = p,vυ* < q.
Lemma 4.8. (cf. [Lin6, Lemma 2]). Let A be a separable simple unital
C*-algebra of real rank zero, stable rank one and with weakly unperforated
K0(A) of countable rank and let x be a normal element in A. For any e > 0
and integer K > 0, there are open subsets 0i,02,...,0 n °f SP(X) such that
OiC)Oj=<b, [U£ = 1 O i ]-=sp(x) 1
Xi G Oi, and there are projections pi G Her (ft), where qi are spectral projections of x in A** corresponding to the open subsets O», such that
x
where y = (1 - ΣtiPiW
- I V +2Σ
=1
λiPi
< € ,
~ Σ£=IΛ),
and for any λ G sp(rr), ίΛere is λ^ such that
dis(λ, λi) < e.
Proo/. We set Δ = {r G 5 : r(l) = 1} (see 4.7).
474
HUAXIN LIN
Without loss of generality, we may assume that \\x\\ < 1. Denote by B the
C*-subalgebra generated by x and 1. Then B = C(X), where X = sp(rc). Let
r G Δ. So r is a quasitrace defined on A (see 4.7). Since B is commutative,
the restriction of r on B is linear. Hence the restriction of r on B gives a state
on B. By the Riesz representation theorem, the state defines a normalized
Borel measure μτ on X. Let D denote the unit disk. For any open subset
O c D , let qo be the spectral projection of x in A** corresponding to the
open subset OΓ\X. The projection q0 is an open projection in A**. Suppose
that h G B(^ C{X)) such that 1 > h > 0 for all t G O Π X and h{t) = 0 for
1//n
all t G X Π O. Then {/ι } forms an approximate identity for Her(go). It is
clear that
μ r ( O Π l ) = li
n
Let {e£} be an approximate identity for Έίeτ(qo) consisting of projections.
Then
μ τ (OnX)=sup{τ(e*)}.
In fact, we have (r is a quasitrace on A)
τ(h1/ke°n) = Tih^eW2*)
< τ{h}lk) < μτ{0 Π X)
and
'
^
< τ(e°n)
for all k and n. Since hι'ke°n -+ e°n, if fc ->• oo, and /ι1/fce^ ^ /i1/*, if n -> ex),
from above equalities and inequalities, we conclude that
Let Δ o denote the countable subset {τn}. For the simplicity, we use the
notation μ* for the measure μTi.
For any e > 0, there is a finite subsets {Ci>C2>— >Cm} of D such that for
any ζ G Z?, there is an integer i such that
|Ci - Cl < β/32
and for any i, there is j Φ i such that
For each i set
APPROXIMATION BY NORMAL ELEMENTS
475
Fix i, for each e/32 < r < e/16, set
Since μk(Di) < 1 and Sr Π 5r/ = 0, if r ^ r', there are only countably many
r in (e/32, e/16) such that
Sr) > 0.
Since the union of countably many countable sets is still countable, we conclude that for each i, there is r^ G (e/32, e/16) such that
μk(Sri)
=0
for all fc.
Now D \ \JSri is a disjoint union of finitely many open sets Oi, O2> •••? O N
such that the diameter of each Oj is < e/4 and
for all k.
Let {e^} be an approximate identity for J5oi5 where Bo{ is the hereditary C*-subalgebra corresponding to the spectral projection qo{ of x in A**
corresponding to the open subset Oi. (Notice that Bo{ has real rank zero,
whence such an approximate identity exists (see [BP])). Then
j = 1,2,... and 2 = 1,2,..., N. Since ^ (i? \ U^O*) = 0,
as n —> oo, j = 1, .2,...,. Since every r E Δ has the form
oo
r =
where otj > 0 and ΣϊjLi aj
=
1?w
e
conclude that
for all r £ Δ, as n -> oo. Since Δ is compact, by Dini's theorem, the
continuous functions Σ i l i e$ ( τ ) defined on Δ converges to the constant
476
HUAXIN LIN
function 1 uniformly on Δ, as n -> oo. Hence we have projections p* € BOi
such that
> Kr
1-
for all i and r G Δ. So, by 4.7, we obtain
The rest of the proof follows from Lemma 4.3.
D
Theorem 4.9. Let A be a separable simple C*-algebra with real rank zero
and stable rank one. If K0(A) is weakly unperforated and of countable rank,
then A has weak (FN). In other words, a normal element x £ A can be
approximated by normal elements in A with finite spectra if and only if
Γ(x) = 0.
Proof Suppose that x E A is a normal element with Γ(x) = 0. Let e be a
positive number. For e/2, let δ and k be as in 3.12. We may assume that
δ < e/4. By applying 4.8 , there are complex numbers λi, λ 2 ,..., λ Λ G sp(rr)
and mutually orthogonal projections Pi,P2> ~-<>Pn € A such that
n
<e/2,
\\x —
where p = Σ"=i Pi and y = (1 - p)x{\ - p),
Ul-p)x-x(l-p)\\<δ,
\pi]>k[l-p], i = l,2,...,n
and for any λ € sp(a ), there is i such that
dis(λ, λi) < e/2.
As in the proof of 4.4, if δ is small enough, by 3.12, there are normal elements
eMk((l-p)A(l-p))
yi
and z e Mk+1((l -p)A(l
with finite spectra contained in sp(a ) such that
||yθyi-z||<€/2.
-p))
APPROXIMATION BY NORMAL ELEMENTS
477
Without loss of generality, we may assume that yλ = Σ7=ι ^%Qu where q{ are
mutually orthogonal projections in Mk((l — p)A(l — p)) such that Σ7=i Qi —
(1 — p) φ ... Θ (1 — p), where the right side has k copies of (1 — p). Since
i = l,2,...,n
\pi}>k[l-pl
there is a partial isometry
v e ( p θ (1 -p) θ ... θ (1 -p))Mk(A)(p®
(1 -p) θ ... θ (1 -p))
such that
K,
and
vv* = (1 - p) 0 ... θ (1 - p)
(there are k copies of (1 — p)), where p\ = v*qiV. Set
u = (1 -p)
@v.
Notice that
2=1
and IA*^U is still normal and has finite spectrum. Now we have
i
(pi ~ P<) ~
u
*zu
<e/2.
2=1
2=1
Therefore
i{Pi -p'i)
-u*zu < e.
2=1
D
Corollary 4.10.
(FN).
All irrational rotation algebras Aθ have the property weak
Proof. It has recently been shown that all such C*-algebras have real rank
zero (see [BKR]). It is known that Aθ is simple and KO(AΘ) = Z + Zθ (see
[Rf]). By [Pt], AQ has stable rank one. So 4.10 follows from 4.9.
D
Now we consider the class of C*-algebras of real rank zero which are
inductive limits of finite direct sums of circle algebras classified recently by
G. A. Elliott ([E112]).
478
HUAXIN LIN
Corollary 4.11. Let A be a C* -algebra of real rank zero which is an
1
inductive limit of finite direct sums of matrix algebras over C ^ ) . If A is
simple and K0(A) has countable rank, then A has weak (FN).
Proof All such algebras have stable rank one and (weakly) unperforated
K0(A).
D
Corollary 4.12. The Bunce-Deddens algebra has weak (FN).
Corollary 4.13. ([Lin6]). Let A be a simple AF-algebra. If K0(A) is of
countable rank, then A has (FN). In particular, every matroid algebra has
(FN).
Proof All AF-algebras have real rank zero, stable rank one and unperforated K0(A). Since Kλ(A) = 0 for all AF-algebras, Γ(x) = 0 for all normal
elements.
D
Corollary 4.14. Let A = lim-+(An,φn) be the C*-algebra inductive limit
of C*-algebras An of the form C(X,Mk(n)), where φn are unital homomorphisms and X is a finite CW complex. If A is simple and of real rank zero,
then A has weak (FN).
Proof As in the proof of [GL, 3.3], A has stable rank one and K0(A)
is weakly unperforated and of finite rank. So this corollary follows from
4.9.
D
5. Normal Elements with One Dimensional Spectra.
Even though we can not give an affirmative answer to question Q for general
C*-algebras with real rank zero at present, we would like to show that for
normal elements with dim sp(x) < 1, the answer to Q is affirmative.
Lemma 5.1. For any e > 0 and integer n, there is δ > 0 such that for any
(unitaϊ) C*-algebra A of real rank zero, selfadjoint elements X\,XΪ, ...,# n € A
and a nonzero projection p € A, if
\\pxi — Xip\\ < δ
and
XiXj = 0,
i = 1,2,..., n and i φ j , then there are selfadjoint elements j/i, y25 -•••> Vn £ pAp
satisfying
\\Vi - pxφW < e and y i y i = 0,
APPROXIMATION BY NORMAL ELEMENTS
479
i = l,2...,n and i φ j .
Proof. We first assume that x{ > 0. Set Zi = pxip. So z{ > 0. Suppose that
\\pxi - Xip\\ < δ,
where δ is a positive number to be determined. We have
<2nδ.
Notice that pAp has real rank zero (see [BP, 2.8]). It follows from [BP, 2.6]
that there is a nonzero projection qι £ pAp such that
1 2
\\(p — 9i)^i|| < (2nδ) ^
z
1 2
and qτ I Ύ2 i I < (2nδ) ^ .
Therefore \\qιZi\\ < (2nδ)ι/2, i = 2,3,...,n. Set yx = q^z^q^ Then
llyi-^ill < II (pWe also have
\\{p-qί)zi{p-q1)-zi\\<2{2nδf'\
i = 2,3, ...,n. Notice that (p — ίi)^(p — q\) G (p — 9i)A(p — #i). We can
then work in (p — q\)A(p — qι). So, by induction (on n), the lemma follows
for the case that Xi > 0. For general selfadjoint elements x^ we notice that
Xi = (xi)+ - (xi)_ and ( ^ ) + ( ^ ) _ = 0 .
D
Lemma 5 2. Let X be a contractible compact subset of the plane which is
homeomorphic to a union of finitely many (compact) straight line segments.
For any e > 0, there exits η > 0, for any (unitaϊ) C*-algebra A of real rank
zero and a normal element x £ A, if p is a projection in A and
\\px - xp\\ < η,
then there is a normal element y G pAp with finite spectrum sp(y) C X and
\\pxp - 2/|| < 2e.
Proof. We may assume that X = U^Lj, where each Li is a compact line
segment and different Li lies in a different line. If n = 1, we may assume
that I c R . Therefore we may assume that x is self adjoint. If δ is small
enough,
sφxp) C (Xφ (Ί R) U {0},
480
HUAXIN LIN
where
Since A has real rank zero, there is a selfadjoint element z G pAp with finite
spectrum sp( z) C (Xe/2 Π R) U {0} such that
\\x-y\\<e/2.
Then we can replace z by a selfadjoint element y G pAp with finite spectrum
P(y) C X such that
s
lk-y||<c.
Now we assume that 5.2 holds for n < k.
We first show that for any σ > 0, there are mutually orthogonal normal
elements Xι E A with sp(xι) C Li (in the respective corner of A) such that
We will prove this by induction on n. Assume that the assertion is true for
n<k. We will show it is true for n = k + 1.
Since X is contractible, there is at least one Liχ such that Liλ Π (Uj^Lj)
has only one point, say ξlm We may also assume that iλ = 1. Set L\ =
Li \ {£i} Let pi be the spectral projection of x (in ^4**) corresponding to
L\ and p^ be the spectral projection of x (in ^4**) corresponding to (L^)'',
where
(£?)' =-teeLfidistfc,&) <σ/64}.
By Brown's interpolation lemma ([Bn2]), there is a projection q £ A such
that
p'i <q<Pi
Note that
||pia?-(pia? + fi(pi-pi))|| <σ/64.
It is then easy to see that
\\qx - xq\\ < σ/32.
Let r : X —> X \ L\ be a retraction. We have
||(1 -q)r(x)-r(x)(l-
q)\\ < σ/16
and
||(1 - g )z(l - ς) - (1 - g)r(ar)(l - q)\\ < σ/8.
APPROXIMATION BY NORMAL ELEMENTS
481
By our first inductive assumption, there is a normal elemet yx E qAq with
sp(yi) C (U^=2Li) such that
\\(l-q)r(x){l-q)-yi\\<σ/8.
Then, by our second inductive assumption, there are mutually orthogonal
normal elements £ 2 ,#3, ...,Xk+i in (1 — q)A(l — q) with s p ^ ) C Li (in the
respective corner), i = 2,3,..., k + 1 such that
JH-l
<σ/4.
i=2
So
< 3σ/8.
We can write pxx = ah + βpu where α and β are complex numbers and h is
a self adjoint element in A**. Since qxq € A, #Λ# £ A. So gxg is normal. If
σ is small enough, we may assume that sp(qxq) C Lλ. Set #! = qxq. Then
Jb+l
< σ.
t=l
This proves the assertion.
Since A has real rank zero, there are mutually orthogonal projections
ei,e2,...,e&+1, selfadjoint elements hi G eiAe^i = 1,2, ...,Λ + 1 and scalars
Qfi, α 2 , . . . , Oίjfc+i a n d /?i, /?2? ---5 /^Λ + i s u c h t h a t α^ = oiihi + A
I f <^ < ί / 2 a n d
77 < ί/2, by Lemma 5.1, there are mutually orthogonal normal elements
y1,y25 J2/n with spectrum sp(yi) C Li (in the respective corner of pAp)
such that
/ n
\
n
< e.
i=l
i=\
Since each e^Aei has real rank zero, there is a normal element y £ A with
finite spectrum sp(y) C X such that
<e/2.
Therefore, if σ is small enough, we have
482
HUAXIN LIN
D
Lemma 5.3 (cf. [Lin5, Lemma 3]). Let X be a compact subset of the
plane with dim X < 1. Suppose that F and Ω are two proper closed subset of
X such that ΩΠF = 0 and the closure ofX\Ωisa compact subset described
in 5.2. Then for any e > 0, there is δ > 0 satisfying the following: for any
unital C*-algebra A with real rank zero, a nonzero projection p £ A and a
normal element x G pAp with sp(rr) = X, if there exist normal elements
y G (1 — p)A(l —p) with sp(y) C F (as an element in (1 — p)A(l —p)) and
z G A with finite spectrum such that
\\x®y-z\\<δ,
then there is a normal element z' G pAp with finite spectrum satisfying
\\x-z'\\<e.
Proof Let / and g be two continuous functions defined on X such that
0 < / < 1,/(C) = 0, if C € F , /(C) = 1 if C 6 Ω; and 0 < g < l9g(Q = 1,
if C G F, g(ζ) = 0, if ζ G Ω and fg = 0. For any δ > 0, by 2.8 (1), there is
0 < η < δ such that for any two normal elements #1, x2 G A with spectra
contained in X if \\xι — α;2|| < η> then
and
\\g(xi)-g(x2)\\<δ.
Now Λve suppose that
ll
We assume that
2=1
where {pi}"=1 is a set of mutually orthogonal projections in A and A* G
λ
a n d r
X, i = l,2,...,n. Set zx = Σ λi €Ω iί>ή *2 = Σλ^nλiP*
=Σ
Since r commutes with z and (1 — p) commutes with x 0 y, we have
/(*)r = r/(z) = r and g{x © y)(l - p ) = (1 ~ί>)g(x Φ y) = (1 - p).
Therefore
APPROXIMATION BY NORMAL ELEMENTS
483
Consequently,
||r-prp||<2*.
If δ < 1/4, by [Eff, A8], there is a projection r' <p such that
\W - r|| < 2δ
and there is a unitary u E A such that
||tx - 1|| < 4ί,
and u*ru = r'.
Thus
Then
\\pz2 - z2p\\ < \\pzι - zλp\\ + \\pz - zp\\
< \\prz1 - zλrp\\ + 2δ < 2(δ + η).
Since
\\u*Z2U-Z2\\ <8ί,
we obtain that
\\p(u*z2u) - {u*z2u)p\\ < 2(δ + η) + 16δ = 18(5 + 2η.
Put pf = p — r'. Since
r'u*z2u = u*z2ur' = 0,
then
\\p'{u*z2u) - (u*z2u)p'\\ < 325 + 2η.
We notice that sp(u*z2u) is a subset of the closure of X\Ω. It follows from 5.2
that, if both η and δ are small enough, there is a normal element y 6 p'Ap'
with finite spectrum sp(y) C (X \ Ω) such that
Therefore, if 5 and η are small enough,
\\x - u*zxu - y\\ < e.
Notice that u*zιu = Σ λ . Ω λi(ϋ*pin) and p' = p — ix*ru.
D
Theorem 5.4. Lei A be a C*-algebra of real rank zero and let x be a normal
element in A with dim sp(α ) < 1. Then x can be approximated by normal
elements in A with finite spectra if and only ifT(x) = 0.
Proof. As in 3.13, we may assume that A has unit. If dim sp(a ) = 0, then
sp(x) is totally disconnected. The conclusion of 5.4 is trivial in this case.
484
HUAXIN LIN
Now we assume that dim sp(#) = 1. It follows from [F, Satz 1; p. 229] that
C(sp(x)) = lim^ooίCί-Xnj^n+i)), where Xn are one dimensional polyhedra. For any η > 0, there exists n and y € φn,oo(C(Xn)), where φniOO is the
map from C(Xn) into C(X), such that
\\x - φn,oo(y)\\ <η
Note that φn,oo(C(Xn)) == C(Xn), where Yn is a compact subset of Xn.
Without loss of generality, we may assume that Xn is a union of finitely
many line segments in the plane. It is also clear that it suffices to show that
every element in φntoo(C(Xn)) can be approximated by normal elements with
finite spectrum. Thus, we reduce the general case to the case that sp(rr) is
a union of finitely many line segments. By 3.13, for any δ > 0, there are
normal elements y G Mk(A) and z £ Mk+ι(A) with finite spectra contained
in sp(a ) such that
||a?Θy-*|| < ί .
Since y has finite spectrum, we may write y = yι Θ y2 such that sp(yi) Π
sp(y2) = 0 and sp(y2) C F, where both F and the closure of S\F satisfy the
description of the contractible compact subset in 5.2. By applying 5.3, we
have a normal element z' with finite spectrum contained in sp(a ) such that
By applying 5.3 again, we finally obtain a normal element z" E A with finite
spectrum contained in sp(x) such that
II*-*Ί<€.
D
Corollary 5.4. Every normal element x in an AF-algebra with dim sp(a ) <
1 can be approximated by normal elements with finite spectra.
6. Applications.
In [E112], George A. Elliott shows that C*-algebras of real rank zero which
are inductive limits of the form
where X{ is homeomorphic to the unit circle or unit interval, can be completely determined by their i^-groups. Conversely, if G is a countable unperforated graded ordered group with Riesz decomposition property, then
APPROXIMATION BY NORMAL ELEMENTS
485
there exists a C*-algebra A of real rank zero which is an inductive limit
of the above form such that (K0(A),Kι(A)) is isomorphic to G (see [E112]
for his definition of unperforated graded ordered group). These remarkable
results also provide the way to construct such C*-algebras with given K*groups. In particular, A is an AF-algebra if and only K\ (A) = 0. Recently, it
was shown that all irrational rotation C*-algebras are in fact in this class of
C*-algebras of real rank zero (see [EE]). As indicated in Section 1, Elliott's
algebras of real rank zero may well include all C*-algebras of real rank zero
which are inductive limits of the form
where each Xιn is a compact subset of the plane. We show in this section
that at least it is true in some special cases.
Theorem 6.1. Let A — lim_+(An,<£n) be a simple C*-algebra of real rank
zero, where each An has the form
where X%n is a compact subset of the plane. Suppose that K0(A) has countable
rank. Then A is an AF-algebra if and only if Kχ(A) = 0.
Proof. Since every AF-algebra A has trivial Kχ(A), we need only to show
the "if" part. From [DNNP], A has stable rank one. It follows from 4.9
that A has (FN).
We now assume that A is unital. For any e > 0, and Xι,x2,.. ,xm € A
there are an integer N and j/i,2/25 •• ?2/m € ΦOO(AN) such that
We will show that there are a finite dimensional C*-subalgebra B C A and
Zi, z2,..., zm G B such that
To save notation, (without loss of generality), we may assume that
where X is a compact subset of the plane. Since φoo(An) is isomorphic to a
C*-algebras with the form C(Y, Mk), where Y is a compact subset of X, we
may simply assume that AN = 0oo(^iv)
486
HUAXIN LIN
Let {βij} be a matrix unit for Mk. Set eij = 1 ® e^ . We view ei>7 G A.
Notice that βnA/vβn = C(X). Let y be a normal element in CUANCU with
s
suc
P(y) = -X" h that y is a generator for enA n en = C(X). It is easy to see
that it is sufficient to show that for any η > 0, there is a normal element
z G eii^-βn with finite spectrum such that
\\y - z\\ < η.
Notice that enAen is simple C*-algebra with real rank zero and, by [Bnl],
A S βnAβn ®/C. So KoienAen) = K0(A) and K^βnAβn) = ϋΓi(il) = °
By 4.9, such z exists. This proves the case that A is unital.
If A is not unital, then A has an approximate identity {en} consisting
of projections (see [BP]). Fixed n. There are integers n(l),n(2), ...,n(fc),...,
such that n(k) < n(k +1) and, for each &, there is a positive element an(k) £
An(k) such that
By [Eff, A], there is a projection qn(k) G An(k) such that
and there is a unitary u G A such that
||ti - 1|| < 4/fc
and u*enu = qnW.
Clearly, qn(k)An(k)qn(k) is isomorphic to a direct sum of finitely many C*algebra each of which is of the form
C(X)®Mm
for some compact subset of the plane X and positive integer m. This implies
that enAen is a unital simple C*-algebra satisfying the conditions of the
theorem. Prom what we have proved for the unital case, we know that
enAen is an AF-algebra. Consequently, A is an AF-algebra.
D
Corollary 6.2. Let A = \iπL+(An,φn) be a simple C*-algebra. Suppose
that each An is isomorphic C(Xn)®Mm(n) for some integer m(n), where Xn
is a contractive compact subset of the plane. Then A is an AF-algebra if and
only if A has real rank zero (or equiυalently, the projections of A separate
the traces (see [BDR])).
Proof. It follows from 4.14 that A has week (FN). Since each Xn is contractive, Kι(An) = 0. Consequently, Kι(A) = 0. Then Theorem 6.1 applies.
D
APPROXIMATION BY NORMAL ELEMENTS
487
The following is due to H. Su. We present here as a simple application of
Theorem 5.3.
Corollary 6.3. ([Su]). Let A be a C*-algebra of real rank zero. Suppose
that A is an inductive limit of finite direct sums of matrix algebras over
one-dimensional spaces, then A is an AF-algebra if and only if Kλ(A) = 0.
Proof By 5.4, every normal element x with sp(#) < 1 can be approximated
by normal elements with finite spectra, if Γ(x) = 0. The proof is the same
as that of 6.1.
D
Remark 6.4. Even if Kχ(A) φ 0, both algebras in 6.1 and 6.3 are inductive
limits of finite direct sums of matrix algebras over circles. This can be done
by applying 4.9 and 5.4 more carefully. We will discuss these elsewhere.
Acknowlegdemerits. This work was done when the author was in the University of Victoria in 1992 and supported by grants from Natural Sciences
and Engineering Research Council of Canada. The author is very grateful to both Professor John Phillips and Ian Putnam for their support and
hospitality. He would also like to thank N.C. Phillips for some conversation.
During this work the author received a grant from National Natural Sciences
Foundation of China.
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Received August 20, 1993 and revised July 25, 1994.
DEPARTMENT O F MATHEMATICS
UNIVERSITY O F O R E G O N
E U G E N E , O R E G O N 97402-1222
E-mail address: lin@bright.uoregon.edu
Peng Lin and Richard Rochberg, Trace ideal criteria for Toeplitz and
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Carlo Pensavalle and Tim Steger, Tensor products with anisotropic principal series representations of free groups
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Albert Jeu-Liang Sheu, The Weyl quantization of Poisson SU{2) .
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Jun Tomiyama, Topological Pull groups and structure of normalizers in
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PACIFIC JOURNAL OF MATHEMATICS
Volume 173
No. 2
April 1996
A mean value inequality with applications to Bergman space operators
PATRICK ROBERT A HERN and Z ELJKO C UCKOVIC
295
H p -estimates of holomorphic division formulas
M ATS A NDERSSON and H ASSE C ARLSSON
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Group structure and maximal division for cubic recursions with a double root
C HRISTIAN J EAN -C LAUDE BALLOT
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The Weil representation and Gauss sums
A NTONIA W ILSON B LUHER
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Duality for the quantum E(2) group
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Cohomology complex projective space with degree one codimension-two fixed submanifolds 387
K ARL H EINZ D OVERMANN and ROBERT D. L ITTLE
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Isoperimetric inequalities for automorphism groups of free groups
A LLEN E. H ATCHER and K AREN VOGTMANN
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Approximation by normal elements with finite spectra in C ∗ -algebras of real rank zero
H UAXIN L IN
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Interpolating Blaschke products
D ONALD E DDY M ARSHALL and A RNE S TRAY
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Interpolating Blaschke products generate H ∞
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Topological full groups and structure of normalizers in transformation group C ∗ -algebras
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